--- pretty_name: "QÆNTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases" language: - en license: other license_name: qaenthrix-component-terms license_link: LICENSE.md tags: - mathematics - linear-algebra - topology - grassmannian - gramian-factorization - reproducible-research viewer: false --- # QÆNTHRIX 3.0.0 **Sharp Continuous Factorization and Optimal Reference Atlases** This repository is a mathematical research artifact collection: a 42-page manuscript, complete proof text for nine core theorems and 29 supporting propositions, executable constructions, exact-arithmetic checks, seeded numerical audits, and reproducible document sources. Download the files to read or reproduce the research. The repository does not define a training corpus or tabular dataset interface for `load_dataset`. The manuscript studies continuous families of finite-dimensional, complex-linear encoders. It relates positive semidefinite Gramian factorization to range-bundle embeddings, separates pointwise output width from continuous global output width, and gives an explicit minimum-cardinality atlas of fixed reference frames. Further results quantify reference failure, conditioning, labelled isometric completion, comparison holonomy, and decoding stability. [Read the manuscript PDF](QAENTHRIX_EVE_Research_Manuscript.pdf) · [Read the proof source](MANUSCRIPT.md) · [Inspect the verification record](VERIFICATION.json) · [Review mathematical provenance](NOVELTY_REVIEW.md) ## Mathematical setting Let $X$ be compact Hausdorff and let $G:X\to\operatorname{Herm}_n^+$ be continuous. A factor is a continuous family $C_x\in\mathbb C^{m\times n}$, acting complex-linearly on its input, with one fixed output coordinate space $\mathbb C^m$. Its uniform Gramian error is $$ \mathcal E(C,G)=\sup_{x\in X}\|C_x^*C_x-G_x\|_{\mathrm{op}}. $$ For the principal family, $P$ ranges over the **entire complex Grassmannian** $\operatorname{Gr}(r,n)$ of rank-$r$ orthogonal projectors, with $1\leq r0$, three different resource questions have the following exact answers: | Requirement | Exact minimum | What is counted | |:--|--:|:--| | Factor at one fixed parameter | $r$ | Complex output coordinates | | One continuous factor over all parameters, in fixed coordinates | $n$ | Complex output coordinates | | A covering atlas of fixed rank-$r$ references, supplying local $r$-row factors | $r(n-r)+1$ | Reference frames | The atlas count is a reference-storage resource, rather than a global output width. Chart indices and transition data are additional resources. Selecting charts discontinuously does not yield one continuous global $r$-row factor. ## Principal results ### Same-width repair below a positive spectral gap Suppose $G_x$ has constant positive rank $r$ and uniform positive gap $\gamma=\inf_x\lambda_r(G_x)>0$. Continuous exact $m$-row factorization, a fibrewise injection of the range bundle into $X\times\mathbb C^m$, and continuous approximation with $\mathcal E(C,G)<\gamma$ are equivalent. For an approximate factor with error at most $\varepsilon<\gamma$, let $P_x$ be the support projector, $A_x=C_xP_x$, and $B_x=A_x^*A_x$. The explicit repair is $$ D_x=A_xB_{x,E}^{-1/2}G_x^{1/2},\qquad E_x=\operatorname{ran}G_x, $$ where the inverse square root acts on $E_x$ and is extended by zero on its orthogonal complement. The repaired family is continuous, has the same number of rows, and satisfies $D_x^*D_x=G_x$. The support correction obeys $$ \|D_x-C_xP_x\|_{\mathrm{op}} \leq\frac{\varepsilon}{\sqrt\gamma+\sqrt{\gamma-\varepsilon}}. $$ No commutation of $G_x$ and $B_x$ is assumed. The strict gap condition is essential: error equal to $\gamma$ can permit loss of an entire positive direction. ### Sharp continuous minimax law For $G_P=\lambda P+\mu(I-P)$, $\lambda>\mu\geq0$, every continuous factor with $m0$, $$ F_W(P)=PW(W^*PW)^{-1/2} $$ is the canonical orthonormal target frame, and $\sqrt\lambda\,F_W(P)^*$ is an exact $r$-row factor of $\lambda P$. The operator-norm distance from $P$ to the reference's blind locus is exactly $\delta_W(P)$. Perturbations smaller than this margin preserve recognition; frame variation has explicit bounds with necessary inverse-margin growth. The minimum number of fixed rank-$r$ references covering $\operatorname{Gr}(r,n)$ is exactly $r(n-r)+1$. Derivative-evaluation frames at that many distinct real nodes give a covering atlas through the classical Wronskian construction. Minimum cardinality is distinct from optimized conditioning. The larger coordinate atlas of $N=\binom nr$ references guarantees $$ \max_j\delta_{W_j}(P)\geq N^{-1/2} \quad\text{for every }P. $$ This is a proved conditioning certificate. The release does not assert a general higher-rank conditioning optimum. ### Labelled completion, transport, and decoding For $k$ positive weights $\alpha_j$ summing to at most one, label Gramians $G_j(P)=\alpha_jP$ admit a visible-plus-reserve isometric completion. Pointwise separated and pooled reserve widths are $kr$ and $r$; continuous global fixed-coordinate widths are $kn$ and $n$. Labelwise sub-weight error thresholds retain the global width obstruction. Canonical subspace comparisons have $U(r)$ cycle holonomy. Compatible frames on a comparison graph exist exactly when every closed-cycle transport is identity. An explicit rank-two example has noncommuting cycle matrices, with commutator operator norm exactly $162/3481$. Repairing each sub-gap label factor gives an isometric encoder whose adjoint decoder amplifies additive output noise by at most one. Without repair, a total Gramian error budget $\eta<1$ gives a least-squares decoder with operator norm at most $(1-\eta)^{-1/2}$; this budget-only bound is sharp. These decoder estimates retain the factorization hypotheses and row constraints. ## Executed verification The included `VERIFICATION.json` records a passing run using Python 3.12.14 and NumPy 2.3.5. Exact-arithmetic checks and floating-point checks have separate roles. | Check family | Recorded scope | |:--|:--| | Exact finite-word audit | 6,270 rational words; 18,738 prefixes, using `fractions.Fraction` | | Exact Wronskian audit | 494 monomial subspaces; 6,124 jet determinants; 64 integer-polynomial subspaces | | Complex finite-word audit | 400 words; 2,573 prefixes; 96 local-unitarity cases | | Arbitrary-rank numerical audit | 224 cases each for repair, reference factors, inside-radius perturbations, Wronskian atlases, coordinate atlases, transport cycles, and noisy decoding | | Additional arbitrary-rank witnesses | 252 nearest-blind projectors; 140 inverse-margin cases; 28 gap-boundary rejections; 28 structured blind families | | Retained rank-one and relational audit | Full counts in the report, including 84,000 synthetic Haar-distributed rays and 4,608 integrated path steps | The arbitrary-rank audit uses seed `30001004`, dimensions $2\leq n\leq8$, and every $1\leq r